Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Convergence of Hermitian Dynamic Mode Decomposition

Nicolas Boullé, Matthew J. Colbrook

TL;DR

This work establishes a rigorous link between data-driven Hermitian DMD and the spectral content of self-adjoint Koopman operators. By enforcing Hermitian structure, the finite-dimensional approximations yield real spectrum and allow a Galerkin interpretation, enabling convergence of spectral measures to those of the underlying operator. The core theoretical contribution is a general convergence theorem for spectral measures of self-adjoint (including unbounded) operators under finite-section projections, together with a resolvent-based analytical framework. The results are complemented by a numerical demonstration on the 2D Schrödinger equation, illustrating accurate recovery of energy spectra and convergence of associated spectral measures, thereby providing theoretical guarantees for spectral computations in data-driven settings.

Abstract

We study the convergence of Hermitian Dynamic Mode Decomposition (DMD) to the spectral properties of self-adjoint Koopman operators. Hermitian DMD is a data-driven method that approximates the Koopman operator associated with an unknown nonlinear dynamical system, using discrete-time snapshots. This approach preserves the self-adjointness of the operator in its finite-dimensional approximations. \rev{We prove that, under suitably broad conditions, the spectral measures corresponding to the eigenvalues and eigenfunctions computed by Hermitian DMD converge to those of the underlying Koopman operator}. This result also applies to skew-Hermitian systems (after multiplication by $i$), applicable to generators of continuous-time measure-preserving systems. Along the way, we establish a general theorem on the convergence of spectral measures for finite sections of self-adjoint operators, including those that are unbounded, which is of independent interest to the wider spectral community. We numerically demonstrate our results by applying them to two-dimensional Schrödinger equations.

On the Convergence of Hermitian Dynamic Mode Decomposition

TL;DR

This work establishes a rigorous link between data-driven Hermitian DMD and the spectral content of self-adjoint Koopman operators. By enforcing Hermitian structure, the finite-dimensional approximations yield real spectrum and allow a Galerkin interpretation, enabling convergence of spectral measures to those of the underlying operator. The core theoretical contribution is a general convergence theorem for spectral measures of self-adjoint (including unbounded) operators under finite-section projections, together with a resolvent-based analytical framework. The results are complemented by a numerical demonstration on the 2D Schrödinger equation, illustrating accurate recovery of energy spectra and convergence of associated spectral measures, thereby providing theoretical guarantees for spectral computations in data-driven settings.

Abstract

We study the convergence of Hermitian Dynamic Mode Decomposition (DMD) to the spectral properties of self-adjoint Koopman operators. Hermitian DMD is a data-driven method that approximates the Koopman operator associated with an unknown nonlinear dynamical system, using discrete-time snapshots. This approach preserves the self-adjointness of the operator in its finite-dimensional approximations. \rev{We prove that, under suitably broad conditions, the spectral measures corresponding to the eigenvalues and eigenfunctions computed by Hermitian DMD converge to those of the underlying Koopman operator}. This result also applies to skew-Hermitian systems (after multiplication by ), applicable to generators of continuous-time measure-preserving systems. Along the way, we establish a general theorem on the convergence of spectral measures for finite sections of self-adjoint operators, including those that are unbounded, which is of independent interest to the wider spectral community. We numerically demonstrate our results by applying them to two-dimensional Schrödinger equations.
Paper Structure (11 sections, 4 theorems, 49 equations, 4 figures, 1 table)

This paper contains 11 sections, 4 theorems, 49 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman operators
  4. Extended dynamic mode decomposition
  5. Spectral measures of self-adjoint Koopman operators
  6. Hermitian Dynamic Mode Decomposition
  7. A general convergence result
  8. Bounded operators - the easy case
  9. Strong convergence of the resolvent
  10. General convergence theorem
  11. Numerical example

Key Result

Lemma 4.1

Consider the above setup and suppose that $\mathcal{L}$ is bounded. Then, for any bounded, continuous, function $\phi$ on $\mathbb{R}$ and for any $v\in\mathcal{H}$, In particular, $\mu_{v,n}$ converges weakly to $\mu_v$ as $n\rightarrow\infty$.

Figures (4)

  • Figure 1: Summary of the idea of Koopman operators. By lifting to a space of observables, we trade a nonlinear finite-dimensional system for a linear infinite-dimensional system.
  • Figure 2: The first $6$ energy eigenstates and eigenvalues of the Schrödinger operator discovered by Hermitian DMD.
  • Figure 3: The first $100$ eigenvalues of the Schrödinger operator discovered by Hermitian DMD along with the exact ones.
  • Figure 4: Visualization of the approximate measures in \ref{['approx_meas']}. For this example, weak convergence in \ref{['thm:convergence_of_hermitianDMD']} means that the positions and heights of the spikes converge. The heights of the spikes, $c_j$, can be thought of as an energy distribution akin to a Fourier transform (but now provided by the spectral theorem).

Theorems & Definitions (10)

  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • proof
  • Theorem 4.6: Convergence of Hermitian DMD
  • Remark 4.7
  • proof : Proof of \ref{['thm:convergence_of_hermitianDMD']}